Vaporization from Clapeyron equation

This equation follows from equation 66, because vaporization occurs at the constant pressure Moreover, the heat of vaporization is related to the slope of the vapor—Hquid saturation curve through the Clapeyron equation ... 

Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine S equation (2-27). 

Better examples of shortcut design methods developed from property data are fractionator tray efficiency, from viscosity " and the Clausius-Clapeyron equation which is useful for approximating vapor pressure at a given temperature if the vapor pressure at a different temperature is known. The reference states that all vapor pressure equations can be traced back to this one. 

Two estimates will be made using vapor pressure data from the CRC Handbook [63] and the integrated form of Clausius-Clapeyron equation ... 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

It is possible, however, to simplify the calculation of the energy transfer by assuming that the vapor phase is always a saturated vapor. O Connor (Ol) has shown that the rate of approach of a superheated vapor to saturated conditions is extremely rapid when the superheated vapor is in direct contact with its liquid phase. If the vapor phase is assumed to be saturated, the temperature of the phase can be calculated from an integrated form of the Clausius-Clapeyron equation instead of from the vapor-phase energy-transfer equation. 

If the vapor-phase temperature is to be evaluated from the Clausius-Clapeyron equation, the pressure in the two-phase tubular contactor must be known at each axial position. This need once again illustrates the necessity of obtaining an understanding of the hydrodynamics of two-phase systems in order to carry out the design of heat-transfer contactors. 

The vapor pressure of the liquid at the surface Pg can be evaluated from an integrated from of the Clausius-Clapeyron equation if the surface temperature Ts is known. 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

From Appendix E, the molar enthalpy of vaporization of mercury at the normal boiling point is 58.6 kJ/mol. Using the Clausius-Clapeyron equation to find the vapor pressure of mercury at 25°C, we have... 

A From Table 13-1 we know that AHvap = 38.0 kJ / mol for methyl alcohol. We now can use the Clausius-Clapeyron equation to determine the vapor pressure at 25.0° C = 298.2 K. 

A rate of reaction usually depends more strongly on temperature than on concentration. Thus, in a first-order (n = 1) reaction, the rate doubles if the concentration is doubled. However, a rate may double if the temperature is raised by only 10 K, in the range, say, from 290 to 300 K. This essentially exponential behavior is analogous to the temperature-dependence of the vapor pressure of a liquid, p, or the equilibrium constant of a reaction, K. In the former case, this is represented approximately by the Clausius-Clapeyron equation,... 

One of the critical issues in vapor pressure methods is the choice of the procedure to calculate the vaporization enthalpy. For instance, consider the vapor pressures of ethanol at several temperatures in the range 309-343 K, obtained with a differential ebulliometer [40]. The simplest way of deriving an enthalpy of vaporization from the curve shown in figure 2.4 is by fitting those data with the integrated form of the Clausius-Clapeyron equation [1] ... 

Any one of Equations (8.14), (8.15), or (8.16) is known as the Clausius-Clapeyron equation and can be used either to obtain AH from known values of the vapor pressure as a function of temperature or to predict vapor pressures of a hquid (or a solid) when the heat of vaporization (or sublimation) and one vapor pressure are known. The same equations also represent the variation in the boiling point of a liquid with changing pressure. 

Chemical potential, chemical equilibrium (Kp, Kc, Kx), Phase equilibrium (1 component), Phase diagrams Vapor pressure equation from Clapeyron eqn,... 

The heats of fusion and vaporization.—From the lowering of the f.p. of cymene and toluene by the soln. of liquid hydrogen chloride, E. Beckmann and P. Wantig14 calculate the heat of fusion of hydrogen chloride as 10 3 cals, per gram of hydrogen bromide, 7 44 cals. and of hydrogen iodide, 413 cals. D. McIntosh and B. D. Steele calculate from Clapeyron and Clausius equation d log pjdT—XjRT 2, for the mol. ht. 

In a very similar way as discussed above for estimating pi from boiling point data, one can treat the vapor pressure curve below the melting point. Again we use the Clausius-Clapeyron equation ... 

Some of these ambiguities can be partially solved using a simple approach recently proposed by Gamier et al. [62], The sublimation pressure of a solid can be estimated using experimental fusion properties and the vaporization enthalpy derived from the equation of state. Using the Clapeyron equation P b can be approximated by ... 

As we have already observed, the vapor-pressure-temperature curve is nonlinear. To reduce this curve to a linear form, a plot of log (p ) versus (1/T) can be made for moderate temperature intervals. The resultant straight line is described by the following expression, which can be derived from the Clausius-Clapeyron equation. 

Critical Tables (7) give values of vapor pressure of 5.0 and 7.5% NaCl solutions over the range of 0° to 110° C. From these data the BPE for a 7.0% solution (50% recovery) at 1 atm. is readily calculated to be 2.34° F. From the ideal solution law (which should apply well to water in dilute solutions) and the Clausius-Clapeyron equation we get... 

Values of the heat of concentration and heat capacity of sea water near room temperature have been measured experimentally. The heat of concentration values compare favorably with those calculated from the vapor pressure data given by Arons and Kientzler by use of the Clapeyron equation. The heat capacity agrees with tne values reported by Cox and Smith. Calculated values for the heat of concentration and boiling point elevation from 77° to 302° F. at salinities up to 9% are presented in both tabular and graphical form. 

The heat of vaporization, AHvap, of a liquid can be obtained either graphically from the slope of a plot of In Pvap versus 1 /T, or algebraically from the Clausius-Clapeyron equation. As derived in Worked Example 10.5,... 

In the most common thermodynamic case, the Clapeyron equation is used with pure components to obtain the heat of vaporization from pure component two-phase (vapor pressure) data. The Clapeyron equation is one of the primary successes of thermodynamics, because it enables the calculation of AH, which is difficult to measure, from easily available properties of pressure and temperature. 

In this case, the pressure increase is due to the vapor pressure of volatile compounds. Often the solvent can be considered as the volatile compound, so its vapor pressure can be obtained from a Clausius-Clapeyron equation ... 

The vapor pressure can be estimated from a Clausius-Clapeyron equation ... 

In general, the molar enthalpy of vaporization is obtained from the Clausius-Clapeyron equation, representing the difference per mole of the enthalpy of the vapour and of the liquid at equilibrium with it ... 

Derive the Clausius-Clapeyron equation [Eq. (44)] from Eq. (40) by neglecting the volume of the condensed phase and using the ideal gas law for the vapor. 

Data were also obtained by this method for the solid states for the methyl ester of 2,4-D, the n-propyl ester of 2,4,5-T, and the butyl ester (liquid) of 2,4-D. The results are shown in Table III. These data were fitted by the least squares method to the Clausius-Clapeyron equations given in footnotes to Table III. These equations were used to estimate the vapor pressures at several temperatures, including the melting point. In Table IV, these are compared with estimates from other sources. Jensen s unpublished data with the Knudsen method compare favorably with those reported in this work, but the published values obtained by other methods are larger. 

The agreement in the case of the published data is not close. It is felt that this is contributed to by the process of extrapolation which is, at best, a hazardous occupation. In many cases such as this the measurements are made at elevated temperatures, and vapor pressures at room temperature are calculated from the Clausius-Clapeyron equation. There are two uncertainties in this extrapolation process uncertainty in the slope of the line of best fit and inaccuracy in the equation itself. 

The deviations observed between extrapolated estimates from GLC data, and direct measurements with the effusion measurements appear to be too large to be accounted for by extrapolation uncertainties. The best estimate can probably be obtained by fitting the combined data to the Clausius-Clapeyron equation (footnote b of Table IV). The obvious implication is that where possible, extrapolation of pesticide vapor pressures obtained at elevated temperatures be converted to interpolation by including a direct measurement at room temperature. In terms of the work described here, vapor pressure measurements requiring the DTA should be supplemented with Knudsen cell measurements. This would require a temperature at which the vapor pressure was 10 3 mm. or less. 

In connection with the orifice test, the vapor pressure of the solid state of this pyrimidine was also determined over a range of temperature from 0° to 50°C. The Clausius-Clapeyron equation fitted by least squares to the 14 measurements made was ... 

From data for Psu at 130 and 140(psia), estimate a value for dP "/dT at I35(psia) and apply the Clapeyron equation to estimate AS1" at 135(psia). How well does this result agree with the steam-table value Apply appropriate generalized correlations for evaluation of VR, HR, and SR for saturated vapor at 13S(psia). How well do these results compare with the values found in (c) ... 

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